Consider the problem of counting the number of ways one can tile a regular hexagon of side N with unit-sized rhombuses. Each rhombus is allowed to be rotated in three ways, depicted with three different colors in the figure on the right. In solving this problem, it is useful to introduce some bijections between tilings, cube stackings in 3D space and vertex disjoint paths on a grid. The purpose of this exposition is to provide an interactive tool for understanding one side of such a bijection.
For example, look at the figure on the right where a hexagon of side N = 4 is tilled with rhombuses. It might be a bit unintuitive at first, but with some imagination one can interpret the picture as showing cubes, stacked on top of each other in 3D space. Specifically, the red tiles correspond to top faces of cubes (or sometimes parts of the "floor" of the room), and the blue and yellow tiles correspond to side faces (or parts of the "walls" of the room). One way to represent a given configuration of stacked cubes is by an NxN matrix of integers denoting the height of each stack. An example of such a matrix is shown on the right corresponding to the particular tiling on the picture.
A given tiling can always be viewed as such a configuration of stacked cubes (and vice-versa) provided that the top cube of each stack is not, even partially, hidden behind others. In the matrix representation, this condition translates to the requirement that the elements of each row and column are in decreasing order.
This tool allows you to edit the matrix of cube stack heights to any other valid configuration and by clicking Draw it will automatically produce the corresponding tiling of the hexagon. Try it out! You can modify the value of N by just adding/removing rows/columns. Just make sure the matrix is still valid, i.e. for an NxN matrix, each entry should be an integer in the interval [0, N] and every row and column should be decreasing in left-to-right and top-to-bottom order respectively.
This project was motivated by Cornell's MATH 6410 "Enumerative Combinatorics" graduate course.
Licenced under CC BY-NC-SA.